Method and apparatus for bore hole directional logging

ABSTRACT

A method and apparatus for borehole directional logging. The apparatus includes first and second coils. The first coil is adapted for rotation about an axis aligned with the longitudinal axis of the borehole. A gimbal mounted magnetic field producing coil is provided for generating a first magnetic field of predetermined direction with respect to the vertical in the space occupied by the first coil whereby an alternating signal is induced therein representative of the inclination angle of the borehole. The second coil is adapted for rotation at the same rate as the first coil while being subjected to a second magnetic field having at least a component of known azimuth direction, thereby generating an alternating signal in the second coil, the phase angle of which, with respect to the first coil signal is representative of the azimuth angle of the borehole. A further embodiment provides computing apparatus for determining the location of a selected segment of the borehole at any depth including mathematical and trigonometric function operators for generating signals representative of the incremental changes of the borehole position and of the corresponding incremental length segments along the borehole. Also included are computing elements for summing the latter signals thereby obtaining the location of the borehole at any depth. The method includes generating first and second signals representative of the borehole inclination and azimuth, respectively, and in response thereto generating signals of the incremental changes in the location of successive segments of the borehole correlated with a signal representation of the length of said segments, and generating signals representative of the borehole location along its length by summing the latter signals.

W as. v 3,691,3 3. Unlteu Dulles eaten;

Armistead [54] METHOD AND APPARATUS FOR BORE HOLE DIRECTIONAL LOGGINGFontaine C. Armistead, Darien, Conn.

[72] Inventor:

Assignee: Texaco lnc.,

Filed: July 17, 1970 Appl. No.: 62,778

Related US. Application Data Division of Ser. No. 727,141, April 30,1968.

New York, N.Y.

US. Cl ..235/l86, 235/150.27 1m. 0. ..G06g 7/22, G06g 7/78 Field ofSearch ..235/186, 61 GM, 151.32;

[56] References Cited UNITED STATES PATENTS [5 7] ABSTRACT A method andapparatus for borehole directional logging. The apparatus includes firstand second coils. The first coil is adapted for rotation about an ax[451 Sept. 12,1972

aligned with the longitudinal axis of the borehole. A gimbal mountedmagnetic field producing coil is provided for generating a firstmagnetic field of predetermined direction with respect to the verticalin the space occupied by the first coil whereby an alternating signal isinduced therein representative of the inclination angle of the borehole.The second coil is adapted for rotation at the same rate as the firstcoil while being subjected to a second magnetic field having at least acomponent of known azimuth direction, thereby generating an alternatingsignal in the second coil, the phase angle of which, with respect to thefirst coil signal is representative of the azimuth angle of theborehole. A further embodiment provides computing apparatus fordetermining the location of a selected segment of the borehole at anydepth including mathematical and trigonometric function operators forgenerating signals representative of the incremental changes of theborehole position and of the corresponding incremental length segmentsalong the borehole. Also included are computing elements for summing thelatter signals thereby obtaining the location of the borehole at anydepth. The method includes generating first and second signalsrepresentative of the borehole inclination and azimuth, respectively,and in response thereto generating signals of the incremental changes inthe location of successive segments of the borehole correlated with asignal representation of the length of said segments, and generatingsignals representative of the borehole location along its length bysumming the latter signals.

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METHOD AND APPARATUS FOR BORE HOLE DIRECTIONAL LOGGING CROSS-REFERENCETO RELATED APPLICATION This is a division of application, Ser. No.727,141, filed Apr. 30, 1968.

BACKGROUND OF INVENTION This invention relates to a method and apparatusfor borehole directional logging and in one particular embodiment itrelates to a method and apparatus for determining the position of aborehole at any given depth.

in drilling for petroleum and for prospecting for other mineraldeposits, it is of great practical importance to have information of theposition of the borehole along its length with respect to the startingpoint thereof on the earths surface. This information is useful for manypurposes, particularly to insure that the borehole remains within apredetermined land area measured at the surface. Other uses includeprospecting wherein core samples taken of the earths strata may becorrelated with the position of the hole, and geological data inaccordance therewith may be prepared to aid in locating petroleum andother mineral deposits.

When a borehole is drilled through various earth formations, it has atendency to deviate from the vertical and accordingly it drifts to theside to an extent which depends upon the angle of borehole drift, ordeviation from the vertical direction thereof, which is also referred toand hereinafter will be referred to as its inclination angle. In orderto determine the position of the borehole it is necessary to know themagnitude of the inclination angle and its direction, that is, thecompass direction or azimuth angle, of the drift with respect to asuitable reference direction, such as, for example, magnetic north. Thisangle will hereinafter be referred to as the azimuth angle.

Various instruments have been proposed to perform all or part of theforegoing measurements, such as those incorporating electromagneticallycoupled hanging pendulums, or coils, which provide information only ofthe angle of inclination of the borehole, and others which combine theaforementioned electromagnetic hanging elements with a single rotatingcoil wired to provide information of the inclination and azimuth anglesof the borehole. Some of the disadvantages of such prior art devices arethat they are rather inaccurate, provide only intermittant informationof inclination and azimuth angles, and in some instances they aredependent upon the assumption that the earths magnetic field remainsconstant in intensity and direction. However, the earths magnetic fieldalong the length of the borehole is often subject to variations ofmagnitude and direction due to variations in the composition of theearths strata and the occasional presence of magnetic materials therein.Frequently, such variations do not acutely manifest themselves and tendto be erroneously recorded by the instrument as deviations of theborehole. A further disadvantage of the aforementioned prior art devicesis that they provide information only of the inclination and azimuthangles of the borehole and not of its actual position in the earth. Todetermine the latter, it is necessary to plot or chart the formerinformation manually at the surface which is time consuming and furtheradds to the inaccuracy of such prior art techniques.

SUMMARY Briefly stated, one aspect of the present invention providesmethod and apparatus for determining the 10- I cation of a bore hole atany depth and which can be carried out with variousembodiments of alogging instrument for the directional logging of a borehole whichincludes a first coil mounted for rotation about an axis substantiallyin alignment with the axis of the borehole. Means are provided forrotating the first coil about said axis. Means are also provided forgenerating a first magnetic field of predetermined direction withrespect to the vertical in at least part of the space occupied by thefirst rotating coil thereby inducing therein a first alternatingelectrical signal representative of the inclination angle of theborehole. Also included in the apparatus is a second coil mounted forrotation at the same rotational rate as the first coil. The second coilis subjected to a second magnetic field with a horizontal component ofknown azimuth direction, whereby a second alternating electrical signalis induced therein, the phase angle of which with respect to the firstcoil signal is representative of the azimuth angle of the borehole withrespect to the magnetic field component of known azimuth direction. Inone embodiment, the second magnetic field is provided by magnetic meanscoupled with a gyrocompass which maintains the magnetic means in a knowncompass alignment. Inanother embodiment, the second magnetic field isthe earths field.

Another aspect of the invention involves a method for logging a boreholewhich comprises the steps of; generating first and second signalsrepresentative, respectively, of the borehole inclination and azimuthangles, at various positions along "the length of the borehole,generating a signal representative of the incremental changes of lengthalong the borehole, and in response to the aforementioned signals and inaccordance with trigonometric relationships, generating further signalsrepresentative of the incremental changes of location of the boreholesegments in correlation with the corresponding incremental lengthsegments in accordance with a convenient coordinate system such as, forexample, cylindrical coordinates, and generating signals representativeof the location of a selected segment of the borehole at any depth,expressed in the aforementioned coordinate system, by summing theaforementioned incremental signals in reference to that depth.

Another aspect of the present invention involves the provision of novelcomputing apparatus in combination with the aforementioned logginginstrument for directional logging of a borehole by generatinginformation of the borehole location in reference to coordinates havingorigin at the starting point of the borehole on the earths surface.Computer input means are provided for receiving the first and secondsignals from the logging instrument, and for generating signalsrepresentative of the incremental borehole length segments through whichthe logging instrument is traversed. The computer includes mathematicaloperators and trigonometric function operators responsive to the azimuthangle, inclination angle, and incremental length signals, for generatingsignals representative of the incremental change of the location of theborehole segments through the aforementioned incremental lengths, andsumming means for summing the latter signals of the incremental changesof location and for providing signal outputs corresponding to thelocation of a selected segment of the borehole at any depth.

In view of the foregoing it is an object of the invention to provide animproved method and apparatus for the directional logging of boreholes.

Another object of the invention is to provide an improved method andapparatus useful with apparatus for the directional logging of boreholesindependent of the magnetic field of the earth.

Another object of the invention is to provide a method and apparatus forthe directional logging of a borehole determining its position in theearth at any depth.

Another object of the invention is to provide embodiments of an improvedborehole logging instrument which generates information of theinclination and azimuth angles of the borehole.

These and other objects, advantages and features of the invention willbe more fully understood by referring to the following description andclaims taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a vertical sectional view ofthe earth traversed by a borehole containing an embodiment of theinstrument of the invention and illustrating in block diagram form asystem employing features of the invention for determining the positionof the borehole at any depth, and its inclination and azimuth angles.

FIG. 2 is a schematic block diagram illustrating an embodiment of ananalog computing system which may be used as the computer shown in FIG.1.

FIG. 3 is a cross-sectional view of one embodiment of the logginginstrument of the invention, incorporating a compass coil and aninclination coil, illustrated in an inclined portion of a borehole. i

FIG. 4 is an oblique view illustrating in further detail the inclinationcoil of the logging instrument shown in FIG. 3.

FIG. 5 is an oblique view illustrating in further detail a portion ofthe instrument in FIG. 3, particularly the Helmholtz coil and itsmounting provisions which may be used for generating a vertical magneticfield intersecting the inclination coil.

FIG. 6 is an oblique view of a three-dimensional coordinate system andtrigonometric construction illustrating the general case of a rotatinginduction coil, which may be used as the inclination coil or the compasscoil, being subjected to a magnetic field of arbitrary direction.

FIG. 7 is an oblique view of the lower portion of a modified version ofthe instrument of FIG. 3 illustrating features of an embodimentincluding a gyro-compass for providing a horizontal magnetic field ofknown azimuth direction intersecting the compass coil.

FIGS. 8 and 8a are respectively a plan view and a sectional elevationillustrating in further detail the magnetic field producing means of theembodiment of FIG. 7.

FIG. 9 is an oblique view of the lower portion of another modifiedversion of the instrument of FIG. 3 illustrating features of anotherembodiment for providing a horizontal magnetic field of known azimuthdirection intersecting the compass coil.

FIGS. 10 and 10a are respectively a plan view and a sectional elevationillustrating in further detail the magnetic field producing means of theembodiment of FIG. 9.

FIG. 11 is a schematic block diagram illustrating an embodiment of ananalog computer for developing a signal corresponding to the azimuthangle of the borehole, which may be used in conjunction with theembodiments of the logging instrument illustrated in FIGS. 7 and 9.

FIG. 12 is an oblique view of a three-dimensional coordinate system andtrigonometric construction useful in practicing the method of theinvention for determining the position of the borehole at any depth.

FIG. 13 is a plan view ofa portion of FIG. 12.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring now to FIG. 3 whichis a cross-sectional view of one embodiment of a logging instrument forcarrying out the present invention illustrated in an inclined portion ofa borehole, a first rotating coil 10, also herein referred to as theinclination coil, is provided containing a number of turns of wireembedded in a suitable potting material and is coupled, by a coupling12, with a constant speed electric motor 11, such as, for example, asynchronous motor. An upper instrument case 13 is provided from whichextends a flange 14 to which is mounted the motor 11. The upperinstrument case is made of a magnetically permeable material therebyshielding the first coil 10 from the earths magnetic field. A lowerinstrument case 15 is provided which is made of a non-magnetic material,such as aluminum or plastic, and which is joined to the upper instrumentcase 13 thereby forming a continuous cylinder therewith. The upper andlower instrument cases 13 and 15, respectively, are ofa circularcylindrical shape having an outside diameter as close to the boreholesize as free axial movement therethrough permits and the combined lengthof the case 13, 15 is sufficient so that alignment of the instrumentcase as a whole with the borehole axis is assured. Also, to assurealignment centralizers may be used on the upper and lower respectiveportions of the case 13, 15, whereby a smaller exterior shape of thecase would be satisfactory. A flange 16 of magnetically permeablematerial extends from the lower portion of the upper instrument case 13.In the flange 16 is mounted a bearing support 17 in which is mounted abearing 18 which rotatably supports a hollow shaft 19 which is in turnfixed to'the lower portion of the inclination coil 10. The motor 11 andthe bearing support 17 are mounted to their respective support flangesl4 and 16 in a manner so that the axis of rotation of the hollow shaft19 is in parallel alignment with the outer surface of the instrumentcase 13, 15, whereby the axis of rotation of the inclination coil 10 isin substantial alignment with the axis of the borehole. A Helmholtz coil40, the details of which are illustrated in FIG. 5, is providedsurrounding the inclination coil and which generates a magnetic fieldparallel to its central axis. The Helmholtz coil 40 is gimbally mountedto the upper instrument case 13 so that it maintains its central axis ina vertical orientation whereby the rotating inclination coil issubjected to magnetic flux lines of a substantially vertical direction.Since the alternating signal provided by the inclination coil may becalibrated with respect to the inclination of the instrument case 13, 15any magnetic field producing means which is maintained in a constantorientation with respect to the vertical may be used. However, since ahomogeneous vertical magnetic field is preferred, a Helmholtz coil is apreferred simple device for obtaining such a field.

When the instrument case 13 is in a vertical orientation that is, whenit is in a borehole having no inclination, the Helmholtz coil 40 alignsitself with the axis of rotation of the inclination coil 10 whereby theinclination coil during its rotation cuts flux lines of the Helmholtzcoil magnetic field in a manner that develops equal and oppositevoltages with the result that no net voltage is induced in theinclination coil. When the instrument case 13 is tilted through anangle, such as it would assume in a borehole having some inclination,the axis of the Helmholtz coil assumes an angle with respect to the axisof rotation of the borehole. Therefore, in its rotation, the inclinationcoil cuts flux lines of the Helmholtz coil magnetic field in a mannerthat develops a net voltage which is related to the angle of inclinationof the borehole. It can be seen that the voltage induced in theinclination coil is related to the sine of the inclination angle thus:

inc)ary ine Sin 1 I) where:

(E,-,, the average value of the alternating voltage induced in theinclination coil,

K a constant which depends on the number of turns, the area, and therotational rate of the inclination coil, the magnetic field strengthgenerated by the Helmholtz coil, and a numerical constant for convertingpeak value to average value, and

(b, the angle between the field of the Helmholtz coil and the axis ofrotation of the inclination coil, which angle can be seen to be the sameas the inclination angle of the borehole.

Since the magnitude of the voltage induced in the inclination coil isused to measure the borehole inclination angle it is best practice tocalibrate the instrument before its use. One way in which this may bedone is to tilt the instrument through a known inclination angle and tomeasure the average AC voltage generated by the inclination coil, andusing this information to solve for K in accordance with the aboverelationship thus:

ine inc)avg]mI 11ml where [(E,-,,,),,,.,,],,,, the average AC voltagemeasured when the instrument is operated at a known inclination anglefor purposes of calibration. A second rotating coil 20, also hereinreferred to as the compass coil, is provided which is similar inconstruction to the inclination coil and also contains a number of turnsof wire. The compass coil 20 is driven by a flexible shaft 21 which isattached to the lower end of the shaft 19 and rotates about a verticalaxis at the same rotational rate as the inclination coil 10. Attached tothe lower portion of the compass coil 20 is a plumb bob 22 for keepingit vertical. The shaft 21 is transversely flexible but rotationallyrigid such as, for example, an automobile speedometer cable, whereby thecompass coil 20 is caused to rotate at the same rate as the inclinationcoil 10 about a vertical axis under the influence of the weight of itsplumb bob 22. The shaft 19 is sufficiently. long so as to pass throughits bearing 18 and extend sufficiently below the flange 16 so that thecompass coil 20 rotates within the lower instrument case 15 where it isexposed to the earths magnetic field. Therefore, there is induced in thecompass coil 20 an alternating electrical signal which passes throughzero each time the plane of the compass 20 is aligned perpendicular tothe north-south magnetic axis. This alternating signal is carried by apair of flexible leads 23a and 23b through the hollow shaft 19 up toslip rings 24 and 25 on the upper portion of the shaft 19. Also on theshaft 19 are slip rings 26 and 27 which are connected to the leads ofthe inclination coil 10. Contacting the aforementioned slip rings arebrushes 28, 29, 30 and 31, respectively, which, for the purpose ofclarity, are illustrated without mounting provisions. It is to beunderstood that the brushes are mounted in an insulated manner to theflange l4 and that they need not be in any particular angularorientation around the shaft 19 since the slip rings 24-27 areelectrically continuous around the shaft. Two pairs of signal wires 32and 33 are connected to the brushes and respectively carry the signalsinduced in the compass coil 20 and the inclination coil 10. The foursignal wires 32, 33 pass through an opening in the flange 14 and aretied together to form a harness 34 which continues upward to the earthssurface or optionally, to electronic equipment integrally mounted withthe logging instrument. Also provided in the wire harness 34 are wires35 to provide power to the motor 11.

It is to be appreciated by one skilled in the art that it is notessential that the compass coil 20 be suspended on a flexible shaft butmight in another embodiment be mounted on a rigid shaft so that its axisof rotation is the same as that of the inclination coil 10. Such anembodiment would provide a single from the compass coil whose strengthwould vary with the inclination of the instrument. This is not a seriousdrawback since it is only the phase and not the amplitude which is usedto determine the azimuth angle of the borehole. However, in such anembodiment the phase angle between the compass coil and inclination coilsignals is not in general equal to the desired azimuth angle but isrelated to it by a mathematical formula, i.e.; equation (7A) or (7B) asexplained below.

In order for the phase relationship or phase angle between the signalsof the compass coil 20 and the inclination coil 10 to be representativeof the azimuth angle of the borehole it is necessary only that the twocoils retain a fixed rotational orientation with respect to each other.A preferred embodiment, however, is one where the two coils are coplanarunder the condition of no borehole inclination, that is when theinstrument case 13, 15 is in a vertical position.

The description of the following steps best illustrates the operation ofthis preferred embodiment. (I) Assume that the instrument and its shaft19 are placed in a start position, for example, with the normal to thecompass coil plane along the north-south direction. Under thiscondition, when there is no inclination of the instrument the compasscoil 20 rotating in the earths magnetic field generates an alternatingvoltage which,

passes through zero every time the coil is parallel to its startposition orientation. Under this condition the inclination coil 10generates no alternating voltage, since its axis of rotation is alignedwith the direction of the Helmholtz coil field, namely, in a verticalorientation, and further since the coil 10 is shielded from the earthsmagnetic field by the upper instrument case 13. (2) When the instrumentis tilted several degrees from the vertical in the direction of magneticnorth the inclination coil in its rotation cuts some of the verticalmagnetic flux lines generated by the Helmholtz coil in a manner suchthat a net alternating voltage is induced in the inclination coil whichpasses through zero at the same time as the alternating voltage inducedin the compass coil. In this instance, the alternating voltages inducedin the two rotating coils are in phase since the borehole inclination isin a magnetic north-south direction. (3) When the instrument is tiltedthe same angle from the vertical as in the preceding example but in adirection perpendicular to the magnetic north direction, the compasscoil generates the same alternating voltage as in the preceding example,but the voltage induced in the inclination coil is out of phase with thecompass coil voltage and passes through zero each time the normal to theinclination coil plane has its horizontal projection aligned with theazimuth direction of the borehole inclination. Therefore, in thisexample, the inclination coil voltage is 90 out of phase with thecompass coil voltage. It can be seen from the above examples that themagnitude of the inclination coil voltage is a measure of the boreholeangle of inclination, and the phase angle between the inclination coiland compass coil voltages is equal to the azimuth angle of the borehole.

Referring now to FIG. 4 which is an oblique view illustrating anembodiment of the inclination coil which may be used in the logginginstrument, the upper and lower portions of the hollow shaft 19 areillustrated broken away. The inclination coil 10 is illustrated as aplurality of windings cast in a suitable potting compound la which forthe purpose of clarity is illustrated as a block of transparent plastic.Fixed to the lower portion of the block a is the hollow shaft 19 whichcarries the signal wires 23a and 23b from the compass coil. These wiresare also potted in the block and are carried to the upper portionthereof. Extending from the upper portion of the block 104 is the upperportion of the shaft 19 which is preferably potted integrally with theblock 100 and made of a suitable insulating material so that the sliprings 24-17 may be integrally cast therewith. The lower pair of sliprings 26 and 27 are connected with the inclination coil 10 by a pair ofleads in the shaft 19. The upper pair of slip rings 24 and 25 areconnected with the compass coil by the leads 23a and 23b which are castin the upper portion of the shaft 19 and in the block 100 and passthrough the hollow lower portion of the shaft 19 to the compass coilbelow.

It is to be appreciated by one skilled in the art that the constructionof the inclination coil and the compass coil may be varied. All that isnecessary is that the coil windings be suitably supported for rotationand the signal leads insulated.

The compass coil 20 shown in FIG. 3 may be similarly embodied as theinclination coil except that attached to its lower portion is the plumbbob 22, and extending from its upper portion are the flexible signalleads 23a, 23b and the flexible shaft 21.

Referring now to FIG. 5 which is an oblique view of a Helmholtz coil andits mounting provisions which may be used in the logging instrument forgenerating a vertical magnetic field, two coils 41 and 42 are shown, inaxial alignment and spaced apart, each of which contains a plurality ofelectrical windings. The coils 41 and 42 are cast in a hollow cylinder40 of a suitable potting compound which for the purpose of clarity isillustrated as a clear plastic material. Also cast in the cylinder 40,in the lower portion thereof, is a weight 43 in the form of a ring. Apair of terminals 44 and 45 are provided in the upper portion of thecylinder 40 to which are respectively connected a pair of flexible leads49 and 50 which are in turn connected to a source of direct currentelectrical power not shown. The coils 41 and 42 are series connected tothe terminals 44 and 45 by wires 46, 47 and 48 cast in the hollowcylinder 40. When a DC voltage is applied to the leads 49 and 50 thecoils 41 and 42 generate a magnetic field in the ho]- low cylinder 40which is parallel to the axis of the cylinder. Extending radially fromthe upper portion of the cylinder 40 are a pair of pivot ends 55 and 56which are diametrically opposed. The cylinder 40 is balanced and alignedso that when it is freely suspended from the pivot ends 55 and 56 itsmagnetic axis assumes a vertical position. A gimbal ring 57 is providedengaging the pivot ends 55 and 56 and which incorporates pivot ends 58and 59 disposed so that the axis between the latter pivot ends isperpendicular to the axis between the pivot ends 55 and 56 so that whenthe gimbal ring is freely suspended from its pivot ends 58 and 59. theHelmholtz coil cylinder 40 has two degrees of pivotal freedom so thatits magnetic axis assumes a vertical position under the influence of theweight 43. The pivot ends 58 and 59 are in turn pivotally mounted in theupper instrument case 13 as illustrated in FIG. 3. It should be notedthat while a simple pivotal and gimbal mount have been disclosed thatany form of mounting the Helmholtz coil may be used which provides itwith the aforementioned degrees of pivotal freedom such as ballbearings, knuckle joints, or the like.

The apparatus discussed above in reference to FIGS. 3, 4 and 5 involvessubjecting the inclination coil to a first magnetic field havingvertical flux lines and further involves subjecting the compass coil tothe earths magnetic field. It will now be shown that in a more generalsense a directional logging apparatus may be provided in accordance withthis invention wherein the two magnetic fields relied upon forgenerating the directional signals may be magnetic fields lying in anyarbitrary direction. It is merely necessary that the direction of thesefields be known.

FIG. 6 illustrates the general case of an induction coil subjected to amagnetic field of arbitrary direction. There is shown a coordinatesystem, x, y, z, fixed with respect to the earth, z being vertical, andx some reference direction in the horizontal plane, eg the horizontalprojection of the earths magnetic field commonly called magnetic north.A coil of area A and n turns is centered at the origin of x, y, z, androtates at an angular velocity about an axis z which coincides with adiameter of the coil. Axis z is directed at an angle (1), from the zaxis, and the plane of zz' is at an angle 6, from the xz plane. Uniformmagnetic induction B lies in any general direction b which is defined bythe angular coordinates (b and 6 The axis x is defined as lying in theb1 plane and being normal to z. The running value of the rotation of thecoil is 74, i.e. the angle which the normal N to the coil has rotatedfrom x. The angle between zand b is given by the equation:

cos simp,cos6,sin,cos6,+sin,sin6,sin sin6 +cos 1 ,osd (3) It should benoted that (b, and 6, change slowly compared to 6; q), and 6 areconstant. The flux of induction through the coil is Flux AB sin cos6'and the induced Emf E is:

which for dldt small compared to d6'ldt, and for uniform d6/dt w, andfor nABw K provides:

E= K sing sin6 4 Equation (4) reduces to equations (5), (6), (8), and(9) for the following cases of special interest. Case I:

ls defined as the case in which the magnetic field B,

is vertical: B, (B,) d), 0 Equation (3) gives 5 (I), and

E,=nAw(B,) sin 4), sin 6', (5) Case II:

ls defined as the case in which the magnetic field B is horizontal: B(B,,),, Q5 17/2, 4 2 0 cos sin 4), cos6,

and

Cos 0: cos cos 6,

w/cos q 1+sin sin 6, (7A) or Sin 0 8111:!

1lcos a tan (7B) CASE III:

Is defined as the case in which the magnetic field B has a verticalcomponent (8 and a horizontal component (8,

E is given by the sum of terms as in Case I and Case m lIl)z 1 I ur): VT Sin2 ,cos 6, sin(a+6,)] 3 CASE IV:

ls defined as the case in which the rotational axis of coil is vertical,4), 0, and by Equation (7A) or (78) a 6,, with the result that Equation(8) reduces to and a= 6,

The azimuth angle of the borehole, 6,, is obtained by measuring thephase angle, a, between the inclination coil signal, E (as in Case I)and the compass coil signal, E In the case of the compass coil havingits axis vertical (as in Case IV), the measured phase angle, a, betweenE and E,,,, is the desired angle 6, directly. In the case of the compasscoil having its axis tilted from the vertical and the coil being exposedto a magnetic field (e.g. the earths) having both vertical andhorizontal components (as in Case III), the measured phase angle, a,between E and E is not uniquely a function of the angle 6,, because ofthe fact that the vertical component of the magnetic field generates onecomponent of E (the first term in Eq. (8)) which is in phase with Ewhile the horizontal component generates another component of E (thesecond term in Eq. (8)) which is out of phase with E, by the angle a. Inorder to obtain a compass coil signal which is uniquely a function of 6,one must eliminate the component of E which is in phase with E,-,,,(i.e. the first term in Eq. (8)). This can be done in either of twoways: (I eliminate the vertical component of the magnetic field whichgenerates this undesired component of E (i.e. operate in the manner ofCase II); or (2) subtract off the undesired component of E electrically,eg by subtracting the fraction (B,,,), /(B,) of E, from E,,,, obtaining:

m III)z/( I)z] 1 m)r V 1 ,sin(a+6',) (I0) In either case the phase anglebetween E and the new E so obtained will be the angle a. This angle atogether with the measured inclination angle (1), gives the desiredazimuth angle 6, by use of Equation (78).

In summary, either one can use a vertical axis compass coil in a fieldthat may have both vertical and horizontal components and obtain 6,directly from the phase angle between E and E or one can use a tiltedaxis compass coil and by eliminating either the vertical component ofthe magnetic field or the component of E which is caused by it obtain 6,indirectly by computation with equation (7B).

From the foregoing it is readily seen that various embodiments of thelogging instrument of this invention may be provided. One suchembodiment is that illustrated in FIGS. 3, 4, and 5, wherein theinclination coil operates according to case I and its signal is inaccordance with Eq. (5) thus:

im: )lnc( I)z 1 Sin In this embodiment the compass coil operatesaccording to Case IV in the earths field and its signal is in accordancewith Eq. (9) thus:

00m )com( E).r 1+ In this instance one can obtain the inclination angle4), from the amplitude of E and the azimuth angle 0, from the phasedifference between E and E,,,,,,.

Referring now to FIG. 7 which is an oblique view of the lower portion ofa modified version of the instrument of FIG. 3 illustrating features ofan embodiment, including a gyro-compass, for providing a horizontalmagnetic field of known azimuth direction intersecting the compass coil,the upper portion of the instrument is not shown and is as describedabove in reference to FIG. 3. Hence, the inclination .coil operates asin Case I above and its signal is in accordance with the aforementionedequation (Ela). The compass coil illustrated in FIG. 7 rotates at thesame rate about the same inclined axis as the inclination coil but isexposed to a horizontal magnetic field of predetermined azimuthdirection. Hence, the compass coil operates as in Case II and its signalis in accordance with Eq. (6) thus:

com )com( e).t V1 dh l sin (a 0,) (E2b) In this embodiment theinclination angle (1:, is obtained directly from the amplitude of theinclination coil signal, and the azimuth angle (1), is obtained bymeasuring the phase difference a between the inclination coil andcompass coil signals and solving Eq. (7b) for 6, in terms of themeasured 41), and a. This solution may be performed by use of the analogcomputing apparatus illustrated in FIG. 11.

The embodiment of FIG. 7 provides a means for generating within thedirection logging instrument a reference magnetic field for the compasscoil, to replace the earths magnetic field in situations where it isdesirable to do so, that is, for example, where the well is cased withsteel pipe such that the earths magnetic field does not penetrate withinthe casing and therefore cannot be sensed by the compass coil, or wherethe direction of the earths magnetic field varies substantially over theborehole length with the result that it cannot be relied upon for aconsistent reference direction.

Specifically, in FIG. 7 a gyro-compass 70 is shown including a yoke 71,a gimbal ring 72 pivotally mounted within the yoke, and an inertia wheel73 rotatably mounted to the gimbal ring 72. The inertia wheel 73 isrotated by a suitable drive means, not shown, incorporated in the gimbalring 72, which rotates the inertia wheel 73 at a suitable gyroscopicangular velocity in accordance with the present art. Fixed to the upperportion of the yoke 71 is a shaft 74 which passes upwardly throughbearings 75 and 76 which are in turn mounted spaced apart'and coincidentwith the axis of the lower instrument case and supported therein in aflange 77. The shaft 74 is axially positioned with respect to thebearings 75 and 76 by retaining rings 78 and 79. Fixed to the upper endof the shaft 74 is an upper yoke 80 within the arms of which ispivotally mounted an outer gimbal ring 81. An inner gimbal ring 82 ispivotally mounted within the outer gimbal ring 81 in a manner such thatits pivotal axis is deposed at a right angle to the pivotal axis of theouter gimbal ring 81'thereby providing the inner gimbal ring 82 withcomplete pivotal freedom. A plumb bob is suspended from the inner gimbalring 82 and maintains it in a constant horizontal orientation. The innergimbal ring 82 is made as a permanent ring magnet and incorporates apair of diametrically opposed pole pieces 83 and 84 on the inner surfacethereof which provide a horizontal magnetic field in the space withinthe inner gimbal ring. The compass coil 20a rotates in this space. Theupper portion of the compass coil 20a is fixed to an extension of theshaft 19 which passes through the bearing 18 and continues upwardlywhere it is connected with the inclination coil 10 and is rotated by themotor 11 as discussed in reference to FIG. 3. In this embodiment thelower instrument case 15 may be optionally made of either a magnetic ora non-magnetic material since the magnetic field strength provided bythe inner gimbal ring 82 far exceeds the strength of the earth s magnetic field. I

The gyro-compass 70 is aligned with some reference direction, e.g. thenorth-south direction, before the instrument is lowered into theborehole. It thereafter maintains this alignment throughout the boreholeso that the magnetic flux lines between the pole pieces 83 and 84maintain the desired azimuth reference direction and horizontalorientation.

FIGS. 8 and 8a illustrate in greater pictorial detail the constructionof the inner gimbal ring 82 assembled with the plumbbob 85.

Referring now to FIG. 9 which is an oblique view of the lower portion ofanother modified version of the instrument of FIG. 3 illustratingfeatures of another embodiment for providing a horizontal magnetic fieldof known azimuth direction intersecting the compass coil, theinclination and compass coil signals of this embodiment are related tothe inclination and azimuth of the borehole in the same manner as arethe signals of the embodiment of FIG. 7. Hence, the inclination andcompass coil signals are respectively in accordance with Equations (Ela)and (E2b). The yoke is provided axially disposed and pivotally mountedwithin the lower instrument case 15. A shaft 121 extends downwardly fromthe yoke 120 and is mounted in bearings 122 and 123 which are in turnmounted in a flange 124 extending from the lower instrument case 15.Pivotally mounted within the arms of the yoke 120 is an outer gimbalring 125 the pivotal axis of which is disposed approximately at a rightangle to the pivotal axis of the yoke 120. Pivotally mounted within theouter gimbal ring 125 is an inner gimbal ring 126 the pivotal axis ofwhich is mutually perpendicular to the pivotal axes of the outer gimbalring 125 and the yoke 120. Hence, the inner gimbal ring 126 is pivotallyfree to assume any position. The inner gimbal ring 126 is made of anonmagnetic material, and integrally mounted thereon are a pair ofdiametrically opposed bar magnets 127 and 128, mounted with oppositepoles facing each other across the inner space of the inner gimbal ring126. The outer gimbal ring 125 and the lower instrument case 15 are madeof a non-magnetic material permitting penetration of the .earthsmagnetic field into the region occupied by the inner gimbal ring 126.The interaction of the earths magnetic field with the outer poles of themagnets 127 and 128 causes the inner gimbal ring 126 in combination withthe yoke 120 to behave as a compass whereby the inner gimbal ring 126 ismaintained in a constant azimuth orientation. Hence, the magnetic fieldbetween the opposed poles in the inner region of the inner gimbal ring126 is maintained in a constant azimuth orientation. Also, the innergimbal ring 126 is maintained in a horizontal position by a plumb bob129 suspended from the lower surface thereof so that the lattermentioned magnetic field is also maintained in a horizontal orientation.The compass coil 20a is rotated within the inner poles of the magnets127, 128 and is subjected to this magnetic field. The upper portion ofthe compass coil 20a is fixed to an extension of the shaft 19 whichpasses through the bearing 18 and continues upwardly where it isconnected with the inclination coil andis rotated by the motor 11 asdiscussed in reference to FIG. 3.

FIGS. 10 and 10a illustrate in greater detail the construction of theinner gimbal ring 126 including the bar magnets 127, 128 and the plumbbob 129.

It is to be appreciated by one skilled in the art that various otherembodiments of the logging instrument of this invention are possible byselecting the inclination and compass coil configurations from any ofthe aforementioned Cases I through IV.

Referring now to FIG. 1 1 which is a schematic block diagramillustrating an embodiment of an analog computer for developing a signalcorresponding to the azimuth angle of the borehole which may be used inconjunction with the embodiments of the logging instrument illustratedin FIGS. 7 and 9, the signals E and E from the logging instrument aretransmitted to a phase meter 118 which measures the phase angle betweenthe signals and provides a DC output signal corresponding to the phaseangle a. The signal from the inclination coil of the instrument is alsotransmitted to a rectifier 115 which converts it to a direct currentsignal which is a measure of sin This signal is in turn transmitted tothe computer 120 of FIG. 1 for computation of the borehole position andalso to an arc sine operator 116 in FIG. 11 which provides an outputsignal proportional to the borehole inclination angle, This signal maybetransmitted to a chart recorder as illustrated in FIG. 1 where it may beread by a human observer. Since the compass coil of the logginginstruments of FIGS. 7 and 9 operates in accordance with Case II thephase angle between the signals E and E must be corrected in accordancewith Equation (78) in order to obtain the borehole azimuth angle. Theoutput signal a of the phase meter 118 is transmitted to a sine operator101 which provides an output signal corresponding to sin a. The a signalis also transmitted to a (cosine) operator 102 which provides an outputsignal corresponding to cos a. The signal proportional to from the arcsine operator 116 is also transmitted to a (tangent) operator 103 whichprovides an output signal corresponding to ten which is in turntransmitted to a multiplication operator 104. The cos a signal from the(cosine) operator 102 is also transmitted to the multiplication operator104 which multiplies its two input signals and adds unity. The outputsignal from the multiplication operator 104 hence corresponds to thevalue:

t4 1 cos a tan% 1 This signal is in turn transmitted to a square rootoperator which computes the square root of this value and transmits asignal corresponding thereto to a division operator 106. The signalcorresponding to sine: from the sine operator 101 is also transmitted tothe division operator 106 which divides the former of its input signalsinto the latter. The output of the division operator 106 thereforecorresponds to sin 0 in accordance with equation (7B). This signal istransmitted to an arc sine operator 107 which provides an output signalcorresponding to 0 the borehole azimuth angle. The 0 signal may then betransmitted to the computer illustrated in FIGS. 1 and 2 for computationof the borehole position, and it may also be transmitted to a chartrecorder where it may be observed.

Referring now to FIGS. 12 and 13: FIG. 12 provides an oblique view of athree dimensional coordinate system useful in practicing the method ofthe invention for determining the position of the borehole at any depth,and FIG. 13 is a two dimensional view, in the xy plane, of a portion ofFIG. 12. The following glossary of terms applies to these figures and tothe descriptions hereinafter included:

(15 the borehole inclination angle.

9, the borehole azimuth angle measured from a reference axis X,,

x,y,z the respective axes of a Cartesian coordinate system having itsorigin on the axis of the borehole at the earths surface with z axisvertical; the x axis being preferably in the direction of eithermagnetic north or geographic north,

x,,y1,z1 the respective axes of a translated Cartesian coordinate systemwith axes parallel to those of the x,y,z system, with origin at anarbitrary point P anywhere along the depth of the borehole,

r the distance in a cylindrical coordinate system measured from the zaxis to any point P along the borehole axis,

1 the angular displacement (in the cylindrical coordinate system) of anypoint P along the borehole axis in reference to the axis x of knowndirection,

L the depth to any point along the path of the borehole,

AL the incremental change of L from one point P" to another point Calong the borehole,

Ar the incremental change of r between the same points corresponding toAL,

An the incremental change of 17 between the same points corresponding toAL.

In order to determine the location of the borehole at any depth it isnecessary to perform an integration, or summing up, of the effect of allof the instantaneous values of inclination and of azimuth angle over theentire borehole path from the earths surface to the given depth L. Thecoordinate system, x,y,z has its origin at the center of the borehole atthe earths surface, with the z axis vertical. At any depth 2, the centerof the borehole has wandered to the point P indicated by the cylindricalcoordinates z, r and 1 Assume that at this point the instrument of theinvention measures the inclination angle to be d), and the azimuth angleto be 0,. The next increment of depth A2 is reflected in a correspondingincrement of length along the borehole AL to the terminal point thereofC. C is at a distance H-Ar from the z axis, and at an angle n+A 17 fromthe x axis measured in a horizontal plane, that is, measured in a planeperpendicular to the z axis. The desired result is to obtain thelocation of the borehole, that is, of any point P along its length,expressed in terms of r and 1; for any known length L along the boreholepath, having continually measured and 0, for each AL along the boreholepath to the point P. Applying the law of cosines to the triangle ABC ofFIG. 13 we have:

By substituting the equations (12) in equation (II) and simplifying wehave:

Ar= AL sind: cos(0,-'n)

and:

L AL sin cos T V 1 1 AL sin sin (1;0

A =arc sin n I: T+AT and for small Ans:

sin (0 -1;)

L 1': (600+: AL Sm Q51 T where:

(0 the value of the angle 6, when r first equals or exceeds 6, theminimum detectable amountof r.

Equations (13) and (15) above provide the cylindrical coordinates, r and1;, of the location of the borehole at any depth L, measured along itspath, after any sequence of values of inclination angle and azimuthangle have occurred and which have been continually measured inreference to every increment of length, AL, along the borehole. It willnow be shown that it is possible to obtain the values of the coordinatesof the location of the borehole by computation, which may be carried outsimultaneously with the running of the directional log, or the data maybe stored and the computation carried out subsequently. In eitherinstance the computational steps may be performed with the aid ofequipment lowered into the borehole in conjunction with the logginginstrument, or the steps may be performed at any time on the earthssurface.

Referring now to FIG. 1 which illustrates a vertical section of theearth traversed by a borehole containing the instrument of the inventionand illustrates in block form a system employing features of theinvention for determining the position of the borehole at any depth, andits inclination and azimuth angles, the logging instrument 13, 15discussed in reference to FIGS. 3, 4, and 5 is illustrated in aninclined portion of the borehole. A logging cable is provided forraising and lowering the logging instrument through the borehole. Thelogging cable 110 contains three conductors 111, 112 and 113, and passesover a conventional wheel or sheave 114, which is mounted for rotation.The conductor 111 carries electrical power down to the instrument. Theconductor 112 carries the alternating signal E from the inclination coilof the instrument, and the conductor 113 carries the alternating signalE from the compass coil of the logging instrument. The inclination coilsignal is transmitted to a rectifier 115 which converts it to a directcurrent signal which is a measure of sin This signal is in turntransmitted to an arc sine operator 116 which provides an output signalwhich is a measure of the angle (b itself. This signal is in turnrecorded by a chart recorder 117 as shown. The inclination coil signalin the conductor 112 is also transmitted to a phase meter 118 to whichis also transmitted the compass coil signal in the conductor 113. Thephase meter 118, when used with the instrument as embodied in FIG. 3,measures the phase difference a between its alternating signal inputsand provides a direct current output signal which is a measure of theazimuth angle This azimuth angle signal is transmitted to the chartrecorder 117 where it is recorded. A suitable phase meter which may beused is that manufactured by Ad-Yu Electronics Inc., Passaic, NewJersey, under the trade designation: Type 405L Precision Phase Meter. Alength measuring and pulse generating unit 119 is provided which ismechanically coupled with the sheave 114 and includes means formeasuring each increment of cable length such as, for example, 1 foot,passing over the sheave 114 as the logging instrument is raised orlowered, and issues a trigger pulse for each such length increment. Thelength measuring unit 119 also includes means for providing a continuouselectrical output signal corresponding to the cable length. This lengthsignal is transmitted to the recorder 117 where it is recorded. Thetrigger pulse representative of the passage of incremental lengthsegments of the logging cable are transmitted to a computer 120. Alsotransmitted to the computer 120 are the direct current signals from therectifier 115 and from the phase meter 118 respectively, of sin (12, andof the azimuth angle The computer 120 is as illustrated in FIG. 2; itsolves equations (13) and (15) for r and 1 and provides correspondingoutput signals which are transmitted to the chart recorder 117 wherethey are recorded.

In summary, it is seen that five signals are concurrently recorded bythe chart recorder 117, to wit: 4),, L, r, and 1 The first three signals0,, and L, provide the inclination and azimuth angles of the borehole atany depth L therein at which the logging instrument is placed. The lastthree signals L, r, and 1;, provide the location of the borehole at anydepth L, in terms of a radius coordinate r and an angle coordinate 1;.

It is to be noted that the quantity 1; appears in the equation 13) for rand that both the quantities 1; and r appear in the equation (15) for1;. As a consequence the equations 13) and (15) must be solved by trialand error using successively determined values of 17 and r. If theequations are solved by digital electronic techniques this problem canbe overcome readily, without introducing any significant error, sincesuch successive calculations may be pre-programmed and rapidlyperformed. In this method the first value of r is computed in accordancewith equation (l3).using for the value of 1 the first perceptible valueof 6 measured in the course of the log. This solution provides a firstvalue of r which together with the assumed value of n, that is, thefirst perceptible value of may be used to compute a first value of 1 inaccordance with equation (15). Subsequent computations are then madeusing the most recently computed values of r and 1 that is, the jthcomputation of r and 1 uses the (j-l)th computed values of r and 1; inthe equations. It can be seen that following several successivesolutions for r and 1 the equations (l3) and (I5) subsequently provideac curate summations determining the values of r and n. It should benoted that erroneous results may be obtained from these computations inthe following situation. Assume that the borehole has at some depthdeparted from true vertical and wandered out to a position having an rvalue of 6 feet at an '1 value of 90 from the direction of north andthat subsequently at a different depth it wanders back to an r valuewhich is zero, or sufficiently close to zero that its value is beyondthe resolution capability of the logging apparatus, and that inreference to both and to intermediate depths, the borehole hasmaintained an 1; value of 90. Let us assume that subsequently, at stillanother depth, the borehole wanders off again at an azimuth angleof 180.If in this circumstance equation is solved continuously for all of thesedepths the computer would retain the value of 90 in the total orintegrated value of 1 for the last mentioned depth notwithstanding thatr had become zero at a previous depth. This problem may be avoided bystarting the computation over again each time r approaches apredetermined value close to the resolution capability of the loggingapparatus by using in equation (15) the last determined value of '1 forthe angle (0,) as defined in equation (15). If digital computingelements are utilized in the computer 120, this problem may be readilysolved by programming the computer to start the solution of equations(13) and 15) over again, to reset r to zero, and to carry over to (0,)the latest computed value of 1; when r decreases to the predeterminedvalue 6.

Referring now to FIG. 2, which illustrates an analog computing systemwhich may be used as the computer shown in FIG. 1, each time anincremental length AL of the logging cable 110 is reeled out, or in, thetrigger pulse transmitted to the computer by the length unit 119 isreceived by a gate pulser 90, which provides a gated pulse of fixedduration in response to each trigger pulse representing each AL movementof the logging instrument through the borehole. A gated pulse istransmitted to a gated diode clamp 91 to which is also transmitted theDC signal proportional to sin 4), from the rectifier 115 of FIG. 1. Thegated diode clamp 91 operates as an electronic switch and in response toeach input pulse generates an output pulse the duration of which isequal to the duration of its input pulse and the amplitude of which isproportional to sin 4b,. This signal is transmitted to a multiplicationoperator 92 to which is also transmitted a DC signal proportional to thecosine of (0 -07) which is obtained from a later described portion ofthe computer. The multiplication operator 92 multiplies its two inputsignals and provides a pulse output signal, the duration of each pulsebeing equal to the duration of each input pulse thereto from the gateddiode clamp 91 and the amplitude of each pulse being equal to theproduct of sin (b, and cosine(01 n). It can be seen from equation (l3)that each of these pulse signals corresponds to each of the Ars whichwhen summed provides the total r or horizontal drift of the boreholefrom its starting point. This signal is transmitted to a summationdevice 93, which provides an output signal corresponding to theaccumulated sum of the amplitudes of its input pulses. The output signalfrom the summation device 93 the refore corresponds to r, the totalhorizontal drift of the borehole from its starting point on the earth'ssurface. It is to be understood that the units of r, thus calculated,are in the same units as those of L. Thus, if AL is 1 foot, the units ofthe computed value of r is in feet.

The DC signal representing 0 the azimuth angle of the borehole istransmitted from the phase meter 118 shown in FIGS. 1 and 11 to asubtraction operator 94 shown in FIG. 2. Also transmitted to thesubtraction operator 94 is a feedback signal from the output section ofthe computer corresponding to n. The subtraction operator 94 subtractsthe former signal from the latter and provides an output DC signalproportional to (O -1 This signal is transmitted to a cosine operator 95and to a sine operator 96. The former provides a signal corresponding tocosine (6 -1 and transmits it to the multiplication operator 92 for usethereof in the computation for r as described above. The sine operator96 provides an output signal corresponding to sin(0 -'r and transmitsthe signal to a multiplication operator 97. Also transmitted to themultiplication operator 97 is the pulse signal from the gated diodeclamp 91 corresponding to sin (1: The multiplication operator 97multiples its input signals and provides a pulsed output signal, thepulse duration of which is equal to the duration of the pulse providedby the gated diode clamp 91, and the amplitude of which equals theproduct of sin (I), and sin(0 -'n). This signal is transmitted to adivision operator 98, to which is also transmitted the DC signalproportional to r from the summation device 93. The division operator 98divides the former input signal by the latter and provides a pulsedoutput signal, the duration of which is equal to the duration of itspulsed input signal and the amplitude of which is equal to the quotientof its input signals. It can be seen from equation (15) that theamplitude of each output pulse of the division operator 98 isproportional to each increment of angle 1;. This signal is transmittedto a summation device 99 which continually adds its input pulses andprovides a DC output signal corresponding to 1 The output of thesummation device 99 corresponds to 1; only if the proper constant ofintegration is also summed in the summation device along with its inputpulses. This integration constant is (0 ),.which is the value 61 haswhen r first equals or exceeds 6 the predetermined reset value of r.

To develop the (0 signal both the mat signals are transmitted to ann-cancel operator 100, which shorts the 1; signal to ground at all timesexcept when r is greater than e. The cancel operator 100 may include anelectronic switch such as a diode clamp which connects the 1 lead toground at all times except when the r signal is greater than thepredetermined reset value which is set into a control element by adiscriminator setting. In addition to the discriminator the controlelement may include a gating element which supplies a control signal tothe aforementioned electronic switch whereby the switch is open at alltimes when r equals or exceeds the value 6 set into the discriminatorand permits the 1; signal to pass through the cancel operator 100uncancelled. When r becomes less than the discriminator setting theelectronic switch closes and cancels the 1; signal to ground therebyresetting the summation device 99 by eliminating therefrom the entireaccumulated value of 17. Following each occasion when the 1 signal iscancelled and then when r grows again to a point where it equals orexceeds 6, the cancel operator 100 applies a single trigger pulse to agate pulser 101 which in response applies a single pulse to a gateddiode clamp 102. The duration of the latter pulse is equal to theduration of the pulses generated by the gate pulser 90 representing AL.Also transmitted to the gated diode clamp 102 is the DC signalrepresenting from the phase meter 118 shown in FIGS. 2 and 11. The gateddiode clamp 102 permits 0 to pass as a pulse the amplitude of which isequal to the 6 signal and the duration of which is equal to the durationof its input pulse. This pulse signal represents the value of (0,) inequation (15) and is transmitted from the gated diode clamp 102 to thesummation device 99 where it is included in the summed output thereofrepresenting 1 It should be noted that the gate pulser 101 applies itspulse of predetermined duration to the gated diode clamp 102 only oncefor each occasion when 1 is cancelled and restored. Therefore, only one(0 pulse is passed by the gated diode clamp 102 for each sequence ofsummation commencing with the restoration of r to a value which equalsor exceeds its minimum predetermined reset value e. From the foregoingwe see that the outputs of the summation devices 93 and 99 representrespectively, r and n, in accordance with equations (l3)and(l5).

It should be noted that the computer of H6. 2 may be used in conjunctionwith any of the embodiments of the logging instrument herein referred toonce the inclination and compass coil signals are corrected to reflectthe actual values of the inclination and azimuth angles in accordancewith the equations discussed in in reference to FIG. 6.

It is to be appreciated that one skilled in the art that while analogcomputation methods have been disclosed the computations may also beperformed quite advantageously by digital computation techniques eitherconcurrently with the taking of the directional logging data orsubsequently. It is also to be appreciated by one skilled in the artthat some or all of the electronic computing equipment may beincorporated within the logging instrument housing which is lowered intothe borehole. In this instance the signals received on the earthssurface would require less manipulations.

While the invention has been described with a certain degree ofparticularity, it can, nevertheless be seen by the examples hereinaboveset forth that many modifications and variations of the invention may bemade without departing from the spirit and scope thereof.

lclaim:

l. A method for determining the location of a borehole at any depth bydetermining at least first and second spatial coordinates of a secondpoint therein at said depth with respect to a reference first pointtherein having known spatial coordinates and wherein said coordinates ofsaid second point therein are given by a corresponding first and secondpredetermined mathematical relationship between the instantaneousinclination and azimuth angles 5, and 0, respectively of said boreholeand the corresponding increments of length AL of said borehole whereinsaid first spatial coordinate of said second point is a distance rmeasured in a horizontal plane between the projections on said plane ofsaid first point of known coordinates and said second point, whereinsaid second spatial coordinate of said second point is an angulardisplacement 1; measured in a horizontal plane between a line ofpredetermined direction through the projection of said first point ofknown coordinates and a line in said plane joining the projections ofsaid first and second points therein, wherein said first mathematicalrelationship is substantially according to the following equation:

Ar AL sin cos(0 1 sin (15 sin (77 01) An=arc sin I:

comprising the steps of:

a. generating a first signal representative of the inclination angle 45of said borehole said first signal being generated with respect to aplurality of points along the length of said borehole, a said pluralityof points being between said first and second points along said boreholelength;

b. generating a second signal representative of the azimuth angle 6 ofsaid borehole said second signal being generated with respect to saidplurality of points along said borehole;

c. generating a third signal representative of the incremental lengthsegments AL of said borehole between said reference first point and saidsecond point, said third signal generating step comprising; generating aseries of third signals each of which represents an incremental lengthsegment along said borehole length between said first and said secondpoint, said series of third signals accumulatively representing thelength of said borehole between said first and second points, whereinsaid first spatial coordinate of said second point is related with saidfirst, second and third signals in accordance with the accumulated sumof a first mathematical relationship, wherein said second spatialcoordinate of said second point is related with said first, second andthird signals in accordance with the accumulated sum of a secondmathematical relationship;

d. generating fourth and fifth signals r and 1 representativerespectively of said first and second spatial coordinates of said secondpoint at any depth, said fourth and fifth signals being generated bycombining said first, second and third signals in accordance with saidfirst and second predetermined mathematical relationships respectively;

e. generating a sixth signal corresponding to the length of saidborehole between said first and second points, and determining saidaccumulated effect of said inclination and azimuth angles over all theincrements of length AL along the borehole, whereby said fourth, fifth,and sixth signals are representative of the location of said secondpoint in said borehole at any depth, wherein said fourth, fifth, andsixth signal generating steps comprise the steps of: ea. generating saidfourth signal representative of said first spatial coordinate of saidsecond point by summing in response to said first, second and thirdsignals substantially according to the following equation:

L T: 2 AL sin 4 cos -1 eb. generating said fifth signal representativeof said second spatial coordinate of said second point by summing inresponse to said first, second and third signals substantially accordingto the following equation:

A n (01))! 2 L SlIl sin (0 1 L=o T said fifth signal therebycorresponding to the net change of said second spatial coordinatebetween said first point of known coordinates and said second point atany depth in said borehole;

ec. generating said sixth signal corresponding to the length of saidborehole between said first and second points by summing substantiallyaccording to the following equation:

L= AL where in the foregoing equations: i

r said first coordinate of second point at any depth in said borehole, nsaid second coordinate of said second point, L the length along theborehole path from said first point to said second point, AL theincremental length segments between adjacent points along said boreholepath in accordance with said third signals, I d), the inclination angleof said borehole determined with respect to said plurality of pointsbetween said first and second points in ac-- cordance with said firstsignals,

f 6 the azimuth angle of said borehole determined with respect to saidplurality of points between said first and second points in accordancewith said second signals;

(0 the value of the angle 0 when r first equals or exceeds the minimumdetectable amount of r. 2. A directional logging system for determiningthe location of a borehole at any depth comprising;

e. means including a logging instrument for generating signalsrepresentative of at least first and second spatial coordinates of asecond point therein at said depth with respect to a reference firstpoint therein having known spatial coordinates;

b. means for traversing said borehole with said logging instrumentbetween said first and second points therein, whereby said signalsrepresentative of at least said first and second spatial coordinates ofsaid second point are generated with respect to a plurality of pointsbetween said first and second points; c. first instrumentation means forgenerating a first signal representative of the inclination angle ofsaid borehole at a plurality of points between said first and secondpoints therein; (1. second instrumentation means for generating a 25second signal representative of the azimuth angle of said borehole ataplurality of points between said first and second points therein; e.length measuring means for measuring the incremental length segments ofsaid borehole between said first and second points therein and forprovid ing a third signal representative thereof, said length measuringmeans being coupled with said means for traversing said borehole andincluding pulse generating means for providing said third signal in theform of a series of pulses said pulses representing the incrementallength segments of said borehole between said first and second pointstherein; f. a computer coupled with said first and secondinstrumentation means and coupled with said length measuring means, saidcomputer comprising: fa. first computing means responsive to said first,

second and third signals for computing the incremental changes of saidfirst spatial coor- 5 dinate occurring between said first and secondpoints in said borehole and for generating a sixth signal correspondingthereto, said first computing means being adapted to compute saidincremental changes in accordance with a predetermined firstmathematical relationship relating said incremental changes with saidfirst, second and third signals; fb. first summing means connected withsaid first computing means for summing said incremental changes bysumming said sixth signal and providing an output fourth signal; fc.second computing means responsive to said first, second and thirdsignals for computing the incremental changes of said second spatialcoordinate occurring between said first and second points in saidborehole and for generating a seventh signal corresponding thereto, saidsecond computing means being adapted to com- 65 pute said incrementalchanges in accordance with a predetermined second mathematicalrelationship relating said incremental changes with said first, secondand third signals; and

1. A method for determining the location of a borehole at any depth bydetermining at least first and second spatial coordinates of a secondpoint therein at said depth with respect to a reference first pointtherein having known spatial coordinates and wherein said coordinates ofsaid second point therein are given by a corresponding first and secondpredetermined mathematical relationship between the instantaneousinclination and azimuth angles phi 1 and theta 1 respectively of saidborehole and the corresponding increments of length Delta L of saidborehole wherein said first spatial coordinate of said second point is adistance r measured in a horizontal plane between the projections onsaid plane of said first point of known coordinates and said secondpoint, wherein said second spatial coordinate of said second point is anangular displacement Eta measured in a horizontal plane between a lineof predetermined direction through the projection of said first point ofknown coordinates and a line in said plane joining the projections ofsaid first and second points therein, wherein said first mathematicalrelationship is substantially according to the following equation: Deltar Delta L sin phi 1 cos( theta 1Eta ) wherein said second mathematicalrelationship is substantially according to the following equation:comprising the steps of: a. generating a first signal representative ofthe inclination angle phi 1 of said borehole said first signal beinggenerated with respect to a plurality of points along the length of saidborehole, a said plurality of points being between said first and secondpoints along said borehole length; b. generating a second signalrepresentative of the azimuth angle theta 1 of said borehole said secondsignal being generated with respect to said plurality of points alongsaid borehole; c. generating a third signal representative of theincremental length segments Delta L of said borehole between saidreference first point and said second point, said third signalgenerating step comprising; generating a series of third signals each ofwhich represents an incremental length segment along said boreholelength between said first and said second point, said series of thirdsignals accumulatively representing the length of said borehole betweensaid first and second points, wherein said first spatial coordinate ofsaid second point is related with said first, second and third signalsin accordance with the accumulated sum of a first mathematicalrelationship, wherein said second spatial coordinate of said secondpoint is related with said first, second and third signals in accordancewith the accumulated sum of a second mathematical relationship; d.generating fourth and fifth signals r and Eta representativerespectively of said first and second spatial coordinates of said secondpoint at any depth, said fourth and fifth signals being generated bYcombining said first, second and third signals in accordance with saidfirst and second predetermined mathematical relationships respectively;e. generating a sixth signal corresponding to the length of saidborehole between said first and second points, and determining saidaccumulated effect of said inclination and azimuth angles over all theincrements of length Delta L along the borehole, whereby said fourth,fifth, and sixth signals are representative of the location of saidsecond point in said borehole at any depth, wherein said fourth, fifth,and sixth signal generating steps comprise the steps of: ea. generatingsaid fourth signal representative of said first spatial coordinate ofsaid second point by summing in response to said first, second and thirdsignals substantially according to the following equation: eb.generating said fifth signal representative of said second spatialcoordinate of said second point by summing in response to said first,second and third signals substantially according to the followingequation: said fifth signal thereby corresponding to the net change ofsaid second spatial coordinate between said first point of knowncoordinates and said second point at any depth in said borehole; ec.generating said sixth signal corresponding to the length of saidborehole between said first and second points by summing substantiallyaccording to the following equation: where in the foregoing equations: rsaid first coordinate of second point at any depth in said borehole, Etasaid second coordinate of said second point, L the length along theborehole path from said first point to said second point, Delta L theincremental length segments between adjacent points along said boreholepath in accordance with said third signals, phi 1 the inclination angleof said borehole determined with respect to said plurality of pointsbetween said first and second points in accordance with said firstsignals, theta 1 the azimuth angle of said borehole determined withrespect to said plurality of points between said first and second pointsin accordance with said second signals; ( theta 1)0 the value of theangle theta 1 when r first equals or exceeds the minimum detectableamount of r.
 2. A directional logging system for determining thelocation of a borehole at any depth comprising; e. means including alogging instrument for generating signals representative of at leastfirst and second spatial coordinates of a second point therein at saiddepth with respect to a reference first point therein having knownspatial coordinates; b. means for traversing said borehole with saidlogging instrument between said first and second points therein, wherebysaid signals representative of at least said first and second spatialcoordinates of said second point are generated with respect to aplurality of points between said first and second points; c. firstinstrumentation means for generating a first signal representative ofthe inclination angle of said borehole at a plurality of points betweensaid first and second points therein; d. second instrumentation meansfor generating a second signal representative of the azimuth angle ofsaid borehole at a plurality of points between said first and secondpoints therein; e. length measuring means for measuring the incrementallength segments of said borehole between said first and second pointstherein and for providing a third signal representative thereof, saidlength measuring means being coupled with said means for traversing saidborehole and including pulse generating means for providing said thirdsignal in the form of a series of pulses said pulses representing theincremental length segments of said borehole between said first andsecond points therEin; f. a computer coupled with said first and secondinstrumentation means and coupled with said length measuring means, saidcomputer comprising: fa. first computing means responsive to said first,second and third signals for computing the incremental changes of saidfirst spatial coordinate occurring between said first and second pointsin said borehole and for generating a sixth signal correspondingthereto, said first computing means being adapted to compute saidincremental changes in accordance with a predetermined firstmathematical relationship relating said incremental changes with saidfirst, second and third signals; fb. first summing means connected withsaid first computing means for summing said incremental changes bysumming said sixth signal and providing an output fourth signal; fc.second computing means responsive to said first, second and thirdsignals for computing the incremental changes of said second spatialcoordinate occurring between said first and second points in saidborehole and for generating a seventh signal corresponding thereto, saidsecond computing means being adapted to compute said incremental changesin accordance with a predetermined second mathematical relationshiprelating said incremental changes with said first, second and thirdsignals; and fd. second summing means connected with said secondcomputing means for summing said incremental changes by summing saidseventh signal and providing an output fifth signal correspondingthereto; whereby said output fourth and fifth signals respectivelycorrespond to the net changes of said first and second spatialcoordinates of said borehole between said first and second pointstherein, said fourth and fifth signals being thereby representative ofsaid first and second spatial coordinates of said second point at anydepth in said borehole.